MSC2010 05C62, 14J80, 37D15
On surfaces glued of $2n$-gons
V. E. Kruglov1, G. N. Talanova2
| Annotation | In this paper $2n$-gons and surfaces obtained through identification of $2n$-gon's sides in pairs (i.e. through sewing) are considered. As well-known, one can get surface of any genus and orientability through sewing but it's very uneasy to calculate by only the polygon and the way of sewing, because to do this one need to calculate the number of vertices appearing after identification; even for small $n$ the problem is almost impossible if one want to do this directly. There are different ways to solve the task. The canonical variant of $4q$-gon sewing ($2q$-gon sewing) giving an orientable (unorientable) surface of genus $q$ is well-known, as the Harer-Zagier' numbers, that are the numbers of variants of sewing a $2n$-gon to an orientable surface of gunes $q$. In this paper we offer a new way of Euler characteristic's of obtained surface calculation (and, hence, its genus) undepending on its orientability by means of three-colour graph and information about closed surfaces topological classification. |
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| Keywords | $2n$-gon, Ejler characteristic, orientability, sewing |
1Vladislav E. Kruglov, Trainee Researcher, Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics (Bolshaya Pecherskaya Ulitsa 25/12, 603155 Nizhniy Novgorod, Russia), student of Institute ITMM, Lobachevsky State University (Prospekt Gagarina (Gagarin Avenue) 23, 603950 Nizhniy Novgorod, Russia), ORCID: http://orcid.org/0000-0003-4661-0288, kruglovslava21@mail.ru
2Galina N. Talanova, student of informatics, mathematics and computer sciences faculty, National Research University Higher School of Economics (Bolshaya Pecherskaya Ulitsa 25/12, 603155 Nizhniy Novgorod, Russia), ORCID: https://orcid.org/0000-0002-4743-4055, Glntalanova@gmail.com
Citation: V. E. Kruglov, G. N. Talanova, "[On surfaces glued of $2n$-gons]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:3 (2017) 31–40 (In Russian)
DOI 10.15507/2079-6900.19.201703.31-40
